6 problems

Let $s_{\mathbb C}(2,4)$ denote the minimum compatibility degree of four dichotomic measurements on a complex qubit. Four-dichotomic-qubit compatibility-degree conjecture. … The va…

Let $k\in\mathbb N$ and $d\geq 2$. The minimum compatibility degree $s_{\mathbb C}(d,2,(2,k+1))$ concerns one dichotomic measurement and one measurement with $k+1$ outcomes on a…

Let $g\geq 2$. For each $d$, let $P_1,\ldots,P_g$ be $g$ independent Haar-random projections of rank $d/2$ on $\mathbb C^d$, and let $A_1,\ldots,A_g$ be their associated dichotomic…

Optimality conjecture. The value $t\leq 1/3$ in the preceding result is optimal: pairs of noisy binary measurements with noise parameter $t>1/3$ cannot, in general, be simulated by…

Nonlinear majorization monotone conjecture. If there exists a blockwise bistochastic matrix $\bm{B}\in\mathcal{B}_{n,d}$ such that $\bm{Q}=\bm{B}\*\bm{P}$, then for any ordering of…

Let $\mathbf{M}=(M_1,\ldots,M_n)$ be an arbitrary extremal rank-one POVM on $\mathbb{C}^d$. A partition $\{X_\gamma\}_{\gamma=1}^\alpha$ of $[n]$ satisfies $|X_\gamma|\leq d-1$ whe…